Cardinal of quotient rings of gaussian integers.

Question

It is known that $\mathbb{Z}[i]$ is a PID and that $\mathbb{Z}[i]/(a+bi)\mathbb{Z}[i]$ is finite for all $(a,b) \in \mathbb{Z}^2\backslash \{(0,0)\}$.

My question :

Is there any result on the cardinal of $\mathbb{Z}[i]/(a+bi)\mathbb{Z}[i]$ ?

Answer 1

Here's a geometric solution: In the complex plane, consider the parallelogram with vertices at $0, a+bi, -b+ai$ and $a-b+(a+b)i$. Show that every point in $\mathbb{Z}[i]$ is congruent to some lattice point in this parallelogram modulo $a+bi$. Now count the number of lattice points that are in the parallelogram using Pick's Formula.

Answer 2

$(a+bi)\Bbb Z$ is the additive subgroup generated by $(a+bi)$ and $i(a+bi) = (-b+ai)$.

So, as a group, $\Bbb Z[i]/(a+ib) \cong \Bbb Z^2/\langle(a,b) ; (-b,a)\rangle$ and its cardinality (up to sign) is $\begin{vmatrix}a & -b \\ b & a\end{vmatrix} = a^2+b^2$.